Problems on Probability

1. Rachel has reached the finals of her tennis club's annual tournament, but she must await the outcome of a match between Linda and Tina before knowing
- who her opponent will be. Observers feel that Rachel has a 50% chance of winning if she plays Linda and a 75% chance of winning if she plays Tina. They also believe the probability that Linda will reach the finals is .8. After the final match is played, you are told that Rachel won. What is the probability that she played Linda? .

2. Assume that a patient is believed to have one of two diseases, denoted by DI and D2 with P(D1) =0.40 and P(D2) = 0.60 and that medical research has determine the probability associated with each symptom that may accompany the diseases. Suppose that given diseases D1 and D2 the probabilities that the patent will have symptom SI, S2, or S3 are as follows:
Symptoms
S1 S2 S3
D1 0.15 0.10 0.15
D2 0.80 0.15 0.03
After a certain symptom is found to be present, the medical diagnosis will be aided by finding the revised probability of each disease. Compute the posterior probability of each disease given the following medical findings:
a. The patient has symptom SI
b. The patient has symptom S2
c. The patient has symptom S3
d. For the patient with symptom S, in part a), suppose we also find symptom S2.
What are the revised probabilities of 0, and 02?
e. If the probabilities for D1 and D2 are 0.25 and 0.75, respectively, compute the revised probability in part d).

3. In 1970 a lottery was held to determine who would be drafted and sent to Vietnam. For each date of the year, a ball was put into an urn. For instance, January I was number 305 and February 14 was number 4. Thus a person born on February 14 would be drafted before a person born on January I. The file data excel 3.xls con¬tains the "draft number" for each date for the 1970 and [97 I lotteries. Do you notice anything unusual about the results of either lottery? What do you think might have caused this result? (Hint: Create a boxplot for each month's numbers.)
Please see data file 3.xls

4. Bin and Irma are planning to take a 2-week vacation in Hawaii, but they can't decide whether to spend 1 week on each of the islands of Maui and Oahu, 2 weeks on Maui, or 2 weeks on Oahu. Agreeing to leave the decision to
change, Bin places two Maui brochures in one envelope, two Oahu brochures in a second envelope, and a brochure from each of the two islands in a third envelope. Irma is to select one envelope, and they will spend 2 weeks on
. Maui if it contains 2 Maui brochures, and so on. After selecting one envelope at random, Irma removes one brochure from the envelope and notes that it is a Maui brochure. What is the probability that the other brochure in the envelope is a Maui brochure?
(HINT: Proceed with caution!)